A bit is a basic unit of information used by conventional computational systems to process information and store information in information-storage devices, such as magnetic and optical discs. The bit is equivalent to a choice between two mutually exclusive alternatives, such as “on” and “off,” and is typically represented by the numbers 0 or 1. Information encoded in bits is called “classical information.” In recent years, advancements in the field of physics have given rise to methods of encoding information in discrete, or continuous, states of quantum systems, including electrons, atoms, and photons of electromagnetic radiation. Information encoded in the discrete states of a quantum system is called “quantum information.” An elementary quantum system has two discrete states and is called a “qubit.” The qubit “basis states” are represented by and and are used to represent the bits 0 and 1, respectively. However, unlike the systems used to realize bits in classical information, such a quantum system can be in the state the state or in a state that simultaneously comprises both and  These qubit states are represented by a linear superposition of states: 
The parameters α and β are complex-valued coefficients satisfying the condition:|α|2+|β|2=1where |α|2 is the probability of measuring the state and |β|2 is the probability of measuring the state 
A qubit can exist in any one of an infinite number of linear superpositions until the qubit is measured. When the qubit is measured in the computational basis  and the qubit is projected into either the state  or the state  The infinite number of qubit-linear superpositions can be geometrically represented by a unit-radius, three-dimensional sphere called a “Bloch sphere”:
          ψ    〉    =                    cos        ⁡                  (                      θ            2                    )                    ⁢                      0        〉              +                  ⅇ                  ⅈ          ⁢                                          ⁢          ϕ                    ⁢                          ⁢              sin        ⁡                  (                      θ            2                    )                    ⁢                      1        〉            where −π/2<θ<π/2 and 0<φ≦π. FIGS. 1A-1C illustrate a Bloch sphere representation of a qubit. In FIG. 1A, lines 101-103 are the orthogonal x, y, and z Cartesian coordinate axes, respectively, and the Bloch sphere 106 is centered at the origin. There are an infinite number of points on the Bloch sphere 106, each point representing a unique linear superposition of the qubit For example, a point 108 on the Bloch sphere 106 represents a qubit  comprising in part the state  and in part the state  However, once the state of the qubit  is measured in the computational basis  and the qubit  is projected into the state  110, in FIG. 1B, or the state  112, in FIG. 1C.
Two or more quantum systems can be used to encode bit strings. For example, the four, two-bits strings “00,” “01,” “10, ” and “11” can be correspondingly encoded in the two-qubit product states  and  where the subscript “1” represents a first qubit system, and the subscript “2” represents a second qubit system. However, the first qubit system and the second qubit system can exist simultaneously in two basis states that are represented by a linear superposition of the product states as follows:
                                        ψ          〉                =                              1                          2                              1                /                2                                              ⁢                      (                                                                              0                  〉                                1                            +                                                                  1                  〉                                1                                      )                    ⁢                      1                          2                              1                /                2                                              ⁢                      (                                                                              0                  〉                                2                            +                                                                  1                  〉                                2                                      )                                                  =                              1            2                    ⁡                      [                                                                                                    0                    〉                                    1                                ⁢                                                                          0                    〉                                    2                                            +                                                                                        0                    〉                                    1                                ⁢                                                                          1                    〉                                    2                                            +                                                                                        1                    〉                                    1                                ⁢                                                                          0                    〉                                    2                                            +                                                                                        1                    〉                                    1                                ⁢                                                                          1                    〉                                    2                                                      ]                              The state indicates that by squaring the coefficient ½ there is a ¼ probability of measuring each of the product states and  when the two qubits are measured separately, each in their computation basis. Certain linear superpositions of the product states, called “entangled states,” can be used in quantum computing and to process and transmit quantum-information. Quantum entanglement is a quantum mechanical property in which the states of two or more quantum systems are linked to one another, even though the quantum systems may be spatially separated. Such entangled states cannot be written as a simple product of a state for each system. The following linear superpositions, called “the Bell states”:
                                                    ψ            -                    〉                12            =                        1                      2                          ⁢                  (                                                                                        0                  〉                                1                            ⁢                                                                  1                  〉                                2                                      -                                                                              1                  〉                                1                            ⁢                                                                  0                  〉                                2                                              )                      ,                  ⁢                                                  ψ            +                    〉                12            =                        1                      2                          ⁢                  (                                                                                        0                  〉                                1                            ⁢                                                                  1                  〉                                2                                      +                                                                              1                  〉                                1                            ⁢                                                                  0                  〉                                2                                              )                      ,                  ⁢                                                  ϕ            -                    〉                12            =                        1                      2                          ⁢                  (                                                                                        0                  〉                                1                            ⁢                                                                  0                  〉                                2                                      -                                                                              1                  〉                                1                            ⁢                                                                  1                  〉                                2                                              )                      ,    and                                        ϕ          +                〉            12        =                  1                  2                    ⁢              (                                                                            0                〉                            1                        ⁢                                                          0                〉                            2                                +                                                                    1                〉                            1                        ⁢                                                          1                〉                            2                                      )            are examples of entangled states. Consider a first qubit system and a second qubit system that have both been prepared in the Bell state  The square of the coefficient 1/√{square root over (2)} indicates that when a measurement is performed to determine the state of the first and second qubit systems, there is a ½ probability of obtaining the result and a ½ probability of obtaining the result Suppose that after the quantum systems have been spatially separated, the first qubit system is measured and determined to be in the state Quantum entanglement ensures that the second qubit system is measured in the state in spite of the fact that the two qubit systems are spatially separated and measured at different times.
Entangled qubit states have a number of different and useful quantum-enhanced applications, such as quantum metrology, quantum cryptography, quantum communication, and quantum teleportation. For the sake of simplicity, quantum teleportation is described below as an example of a quantum-enhanced application. Quantum teleportation can be used to transmit quantum information in the absence of a quantum communications channel linking the sender of the quantum information to the recipient of the quantum information. FIG. 2 illustrates an example of quantum teleportation. In FIG. 2, quantum communications channels are represented by directional arrows, such as directional arrow 202, and a classical communications channel is represented by a dashed-line directional arrow 204. Bob receives a qubit  that he needs to transmit to Alice, but Bob does not know the value of the parameters α and β, and Bob can only transmit classical information over the communications channel 204. Bob can perform a measurement on the qubit, but by the properties of quantum mechanics, a measurement irrevocably destroys most of the qubit information rendering any information that Alice may choose to obtain from  irretrievable. In order for Alice to receive all of the information contained in the qubit both Bob and Alice agree in advance to share a Bell state generated by an entangled state source 206:=1/√{square root over (2)}(|0>AB)where the subscript “A” identifies qubit basis states transmitted to Alice, and the subscript “B” identifies qubit basis states transmitted to Bob.
The overall state the system in FIG. 2 is:=1/√{square root over (2)}(α|0>C+β|1>C)(|0>A|1>B−0>B)where the qubit  is represented by  The state  can be rewritten in terms of the four Bell states as follows:
                  Θ      〉        ABC    =            1      2        ⁡          [                                                                                ψ                -                            〉                        BC                    ⁢                      (                                          α                ⁢                                                                          0                    〉                                    A                                            -                              β                ⁢                                                                          1                    〉                                    A                                                      )                          +                                                                          ψ                +                            〉                        BC                    ⁢                      (                                                            -                  α                                ⁢                                                                          0                    〉                                    A                                            -                              β                ⁢                                                                          1                    〉                                    A                                                      )                          +                                                                          ϕ                -                            〉                        BC                    ⁢                      (                                          β                ⁢                                                                          0                    〉                                    A                                            -                              α                ⁢                                                                          1                    〉                                    A                                                      )                          +                                                                          ϕ                +                            〉                        BC                    ⁢                      (                                                            -                  β                                ⁢                                                                          0                    〉                                    A                                            -                              α                ⁢                                                                          1                    〉                                    A                                                      )                              ]      The state reveals that the Bell states are entangled with the qubits identified by the subscript “A.” Bob is in possession of the Bell states identified by the subscript “BC,” and Alice is in possession of the qubits identified by the subscript “A,” but Alice does not know which of the four qubit states she possesses. Bob and Alice both agree in advance that the strings “00,” “01,” “10,” and “11” correspond to the entangled states  and  so that when Bob performs a Bell state measurement to determine the Bell states in his possession, he can immediately transmit to Alice the corresponding two-bit string over the communications channel 204. As a result, Alice knows which qubit state she possesses. For example, suppose Bob performs a Bells state measurement that outputs the state  Bob's measurement projects the state into the state (α|0>A−β). Bob then transmits the string “01” over the communications channel 204 to Alice. Quantum entanglement ensures that Alice knows with certainty that she possesses the qubit state  which is equivalent to the original qubit state as the overall phase is unimportant. For the other Bell state measurement outcomes, all of which occur with probability ¼, Alice performs operations on the qubit in order to transform the state into the original unknown state supplied by Bob.
Methods used to generate entangled qubits often employ direct interactions between the qubits being entangled, are limited to either matter-based qubits or photon-based qubits, or necessitate performing numerous measurements. As a result, physicists, computer scientists, and users of quantum information have recognized a need for new methods that can be used to generate entangled qubits in both matter-based and photon-based qubits using a single measurement and indirect interactions between qubits.